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Morita equivalence for continuous groups BY IEKE MOERDIJK Mathematical Institute, University of Amsterdam
(Received 25 April 1987)
This paper is essentially concerned with the following problem. Let G be a topological group. A G-set is a set S equipped with a continuous action S xG—*S (where S is given the discrete topology). These G-sets form a category BG. It is well known that if G and H are discrete groups, BG is equivalent to BH iff G is isomorphic to H. However, this result cannot be extended much beyond the discrete case. The problem is: when is BG equivalent to BH, for topological groups G and HI It is of some interest to compare this with the results on Morita equivalence for commutative rings [14]. The analogous problem is: given rings (commutative, with 1) R and S, and an equivalence between their categories of modules: Mod (R) m Mod (S), what is the relation between R and SI An essential ingredient here is that any functor Mod [R)->Mod(S) which preserves co-limits is of the form — ®RP, for a i?-5-bimodule P. (For an exposition, see, e.g. chapter n of [2].) One of the problems in the context of BG is that, unlike Mod (R), BG does not have a single generator. However, for every open subgroup U <=G, the set of right cosets G/U is an object of BG, and the objects of this form collectively generate BG. So if F: BG^-BH is a functor which preserves co-limits, one may construct the space P = lim F(G/U), (1) '— u where the inverse limit is taken over all open subgroups U, ordered by inclusion. Since H acts on each set F(G/U) on the right, there is a continuous action of H on P. To proceed, it turns out that G is somewhat too small: let M(G) = lim G/U. (2) <—u M(G) is a topological monoid, with multiplication defined by
(x-y)u = xu-yx-iUXu where x = (Ux^y, y = (Uy^y are points of M(G). M(G) acts on P from the left,
where x = (Uxv)veM(G),p = (p^ueP and xu: G/xJUxy -+ G/U stands for the map
If S is a G-set, the action SxG-+S can uniquely be extended to a continuous action SxM(G)^-S, since for seS and x = (Ux^ueMiG), the sequence («•&„)„ eventually becomes constant. Analogously to the module case, one can then show that if F preserves co-limits and finite limits,
(3) PSP 103 98 IEKE MOERDIJK i.e. that for each G-set S, F(S) is isomorphic to the quotient of SxP obtained by identifying (s-x,p) and (s,x-p) for seS, peP, xeM(G) (see 3-4 of this paper). There is one necessary condition, though: the inverse limit (1) may be very badly behaved if G is too big. However, if G has a countable neighbourhood basis at the unit-element, for instance, then (3) will follow. This size condition on G can be-eliminated if one takes the inverse limits (1) and (2) in a slightly different category of generalized spaces, or locales (as discussed in [5, 8, 9], and others). Having taken this step, one may as well consider the problem of when BG and BH are equivalent categories for group objects in this larger category of generalized spaces, so-called continuous groups. Another reason for working with continuous groups, rather than just topological groups, is a theorem due to A. Joyal and M. Tierney, which states that any connected atomic topos with a point is equivalent to a category of the form BG, which is Galois theory, generalized from Galois categories in the sense of Grothendieck [4] to arbitrary connected atomic toposes with a point. As a consequence, our results will imply a description of the category of pointed connected atomic toposes and geometric morphisms, purely in terms of continuous groups and certain ' bimodules'; see Theorem 4-4 below. These results are valid over an arbitrary base topos.
1. The classifying topos of a continuous group Throughout this paper, Sf denotes an arbitrary but fixed base topos; we will abuse the language in the usual way and act as if £P is 'the' category of sets. 1-1. Spaces and locales. We will follow the terminology of [9]. So a locale is a complete lattice A which satisfies the distributivity law
a A V bt = \/ a A bt.
A morphism of locales A -»_B is a function which preserves finite meets and arbitrary sups. This defines the category of locales. Its dual is the category of spaces. If X is a space, we write &(X) for the corresponding locale, the elements of which are called opens of X. So a map of spaces X > Y is by definition a locale morphism f-1: d)(Y)^@(X). Cts(X, Y) denotes the set of maps from X to Y. We will assume that the reader is familiar with the basic properties of locales and spaces; see [8, 9]. Change-of-base techniques play an important role in the sequel, and some familiarity with the methods (rather than the results) as exemplified in [13] will be very helpful; cf. also [12], in particular 53. In applying change-of-base techniques in the context of quotients of locales, it is necessary that these quotients be stable. To this end, we make the following important observation.
1-2. LEMMA. Let XZ$ Y >Z be a coequalizer of spaces. If f and g are open, then g so is q. Moreover, in this case IX f TxX Z^TxY-^TxZ 1X0 is a coequalizer, for any space T. Morita equivalence for continuous groups 99 Proof. The coequalizer Z can be constructed as 0(Z) = {U e &(Y) \f~*{U) = g and q~l: &(Z) -> &(Y) is the inclusion. Clearly if/"1 and g~x preserve arbitrary infima l\ and implications =>, so does q'1. So q is open if/ and g are. Now take a space T. If/is open, so is 1 xf: TxX-+T x Y, and if/,: is the left adjoint of/"1, then 1 ®/, is left adjoint to 1 ®f~x,
0(X) "Z Q{T) ® 0( Y)
(recall that ff(TxI) = 0{T) ® 0(X), etc.; cf. [9]), and similarly for g. T pt g {U Clearly 1 ® q~x: fi?(T) ® 0(Z) -> &(T) ® (P(7) factors through g. We claim that S is precisely the image of 1 ® q~x. To see this, take Ue£. For Ve(9(T), let
Then U = \/{V®Wv\Ve0(Y)}. Now take 7e(P(y). Then
1 an( 1 1 So ^!/~ (^) ^ W^K> i hence by adjointness,/" ^^) < g~ (Wv). By symmetry, we l 1 conclude th&t f~ (Wv) = g~ (Wv), i.e. Wve0(Z). Consequently, U is in the image of 1 ® q'1. This shows that S S (9(T) ® 0(Z), and hence that T xl ^T xY^TxZ is a coequalizer.
13. Continuous groups. A continuous group is by definition a group object in the category of spaces. In this paper, we will always assume that G is open, i.e. that G->1 is an open map of spaces. If G is a continuous group and U is an open subgroup, then the quotient G/U defined as the coequalizer
GxUzXG-^G/U (1)
i is discrete. Moreover, (1) is a stable coequalizer and pu is an open surjection, by 1-2. A continuous monoid is a space M equipped with an associative multiplication MxM^-M which has a two-sided unit 1 >M. The following construction plays a fundamental role in this paper. Let G be a continuous group, and let M(G) = lim G/U. (2) '—u be the inverse limit in the category of spaces, taken over all open subgroups [/£(}, partially ordered by inclusion. If U £ V, the projection G/U -+G/V is a surjection of sets, so it follows that the projection 100 IEKE MOERDIJK is an open surjection (surjectivity follows from [6]; that n^ is also open is a special case of theorem 5-l(ii) from [10]).
1-4. LEMMA. The construction oj M(G) is stable; i.e. if 2F >$ is a geometric morphism and G is a continuous group in 1-5. LEMMA. Let Gbea continuous group. Then M(G) has the structure of a continuous monoid, and the canonical map defined by nuo6 = pv preserves the multiplication. Proof, (a) Let us first consider the classical case of a topological group G in sets, and define M(G) = \imG/U where the inverse limit is taken in topological spaces (rather than 'generalized' spaces). Writing the elements of M(G) as sequence x = {Ux^u of cosets (XfjGG), we can define an associative multiplication by (x-y)u = xu-yx-iVxo, (1) and it is easy to check that this makes M{G) into a topological monoid. (b) The same formula (1) works in the context of a continuous group in the base topos Sf, by the method of test spaces and change-of-base. Indeed, to define a multiplication M(G) xM(G)->M(G), we may equivalently define an operation Cts (T,M(G)) x Cts (T,M(G)) -ZZ-*Cts (T,M(0)), X natural in the test space T. So suppose we are given maps TZXM(G), and consider the pullback v r iflZiL *nv{GxG) !Lr Tlo(G/VxG/U) where the product IIt7(...) is taken over all open subgroups U <^G, while p and n are the obvious maps induced by the projections G Vu>G/U, M{G) "u >G/U. Since each pv is an open surjection, so is p, and hence t is an open surjection. Let Sh (T') —?-* Sf be the canonical geometric morphism. Then x' and y' define points of y*(J\.uG) = ftu(y*G) in Sh(T'), and formula (1) defines a point x'-y' of M(y*G) = y*{M{G)), because open subgroups of the form y*(U) are co-final among open subgroups of y*(G), cf. 1-4. Since t is an open surjection, it is a coequalizer of its kernel-pair ([10], p. 66), and it is not hard to see that this implies that x' • y' factors through t, so as to produce the required map T^+M(G). Further details are routine. Morita equivalence for continuous groups 101 1-6. Remark. This type of change-of-base argument can essentially be reduced to the slogan that we may use point-set definitions (like (1)) and arguments in the context of stable constructions of generalized spaces. In the sequel, we will often (implicitly) refer to this slogan, and suppress further details. 1-7. G-sets. Let G be a continuous group. A G-set is a set S (an object of Sf), equipped with a map of spaces SxG-+S satisfying the usual identities for an action; here S is regarded as a discrete space. With the obvious notion of a map of G-sets, we obtain the category BG, called the classifying topos of G. BG is an atomic topos (see [1] for information about atomic toposes). The full subcategory oi BG consisting of the G-sets G/U (with action induced by the multiplication of G) where U ranges over the open subgroups of G, forms an atomic site for BG which we denote by §(G). If U and V are open subgroups of G and x: 1 -» G is a point of G such that U £ x~xVx, then x defines a map G/U->G/V in §(G) by Ug^-Vxg (point-set notation). We simply write G/U >G/V for this map. The construction works over any base topos: if G is a continuous group in a topos $, the topos of objects of $ equipped with an action by G is denoted by B(S, G). The canonical geometric morphism is given by y%(E) = E with the trivial (?-action. yG has a canonical section whose inverse image is the forgetful functor. The construction is stable under change of base (recall that we assume G -*• 1 to be open). 1-8. LEMMA. lf!F -+£ is a geometric morphism and G is a continuous group in S, then there is a natural equivalence Proof. See [11] or [12]. The following theorem is due to Joyal and Tierney[9]. - l 9. THEOREM. Let 3F >$ be a connected atomic geometric morphism. If f has a section p, then there is a continuous group G in & and an equivalence 9 ^— B(S,G) (the section p corresponds to the canonical section pG under the equivalence). This is a representation theorem for pointed atomic connected toposes (over any base, & in this case). The aim of this paper is to investigate how geometric morphisms between pointed connected atomic toposes can be described in terms of the corresponding groups. 102 IEKE MOERDIJK 110. Remark. Let G be a continuous group, and £f PG > BG be the canonical point of the topos BG. Then M(G) is the space of endomorphisms of pG, in the sense Sh(M(G)) is a lax pullback of toposes. Obviously, the definition of the category BG also makes sense if G is just a continuous monoid, and we extend the notation to this context. (If M is a continuous monoid, the corresponding topos BM is in general no longer atomic.) I'll. LEMMA. Let G be a continuous group. The canonical map G • M(G) induces an equivalence of categories Proof. Let S be a G-set. Density of 6 easily implies that the action SxG^S has a unique factorization through Sxd: 1*12. Example. It follows that if G and H are continuous groups such that M(G) and M(H) are isomorphic, then BG is equivalent to BH. The converse is false. For instance, if S is an infinite set, M(Aut (S)) is the monoid of monomorphisms 8 >—> S (constructed as a locale), and B Aut(S) classifies infinite decidable sets (and monomorphisms between them). So B Aut (S) is equivalent to B Aut (8') for any two infinite sets 8 and S'. 2. Bispaces Let G be a continuous group (in a topos Sf), and let M(G) be the associated monoid. An M(G) -space is a space P with an action M(G)xP—'-*P. Such an J/(Cr)-space P is called open if P is an open space (i.e. P-> 1 is open) and the action M(G) xP—'—>P is an open map. If X is a space with an action X xM(G) —'—* X ofM(G) on the right, one can form as e the tensor-product X ®M {) (1) where a. = Xx •, ft — • xP. 2-1. LEMMA. Let P be an open M(G)-space, and suppose that the diagonal action is open (here /i(} is the composite M(G) xM(G) x P "" >M(G) x P —'—*P). Let S be aG- Morita equivalence for continuous groups 103 set, regarded as a discrete M(G)-space (cf. I'll). Then S®M(G)P is discrete, and the coequalizer a is stable. Proof. — ®M(G) P commutes with sums, so by writing S as the sum of its orbits, we may restrict ourselves to the case where S = G/U for some open subgroup U ^ G. Let Nv be the corresponding basic open of M(G), i.e. I I" Nv> is a pullback. Then G/U ®M(G)P is the same as the coequalizer NuxNuxP^P^G/U®mG)P, (3) where fi(j is multiplication of the ith and jth factor. Note that q is open since /i13 and fi12 are (cf. 1-2). Since JP->1 is open, so is G/U®MiG)P^ 1. Moreover, G/U ®U{G)P has an open diagonal, as follows by considering the diagram (with r = /*yog) (/W ) NvxNvxP " ,PxP \ ,x, (4) -^ (G/U®M(G)P) x (G/U®mG)P). Indeed, qxq is open since q is, and (/M13,/M23) is open by assumption. Since r is IS surjective, it follows that A must be open. Consequently, G/U ®M(G)P discrete (cf. [9], §iv.2). Finally, it follows from 1*2 that the coequalizer (2) is stable. 2-2. Notation. Note that G/U ®M(G)P can also be constructed as the coequalizer i . (5) Therefore, we will also write P/Nv for G/U ®M(G)P. An ilf(6r)-space is called flat if it is an open M(G)-space such that the diagonal action map M(G) xM(G) xP-+PxP from 21 is open, and the induced functor (by is left-exact. Notice that since —®MiG)P obviously commutes with co-limits, a flat -M(6r)-space gives rise to a point of the topos BG. 104 IEKE MOERDIJK 2-3. Bispaces. Let G and H be continuous groups. An M(6?)-J/(//)-bispace is an open il/(6r)-space P equipped with another open action PxM(H)—:L-^P of M(H) on the right, such that the two actions are associative, i.e. M{G)X M(G) xPx M(H) \ M(G) x P P x Jf(fl) -P commutes. An M(G)-M(H)-bisipa,ce is called flat if it is flat as an ilf(C?)-space. A map of ilf(G!)-.M(//)-bispaces P *P' is a map of spaces which commutes with both actions. This defines a category Flit (M(G),M(H)) of flat M(£)-il/(#)-bispaces. By 21 and 1-11, each flat M(G)-M(H)-STpace P induces a functor — where for toposes & and 3F over Sf, Homy (IF, S) denotes the category whose objects are geometric morphisms !F *i over SS, and whose maps /=>/' are natural transformations/*-*/'* (over if). If K is another continuous group, P is an M(6r)-ilf(//)-bispace and # is an M(H)- M(^T)-bispace, one can construct the coequalizer PX* PQP. (1) This coequalizer is stable by 1*2, so P ®M(//) © inherits an action of M(G) on the left and one from M(K) on the right, in the obvious way. 2-4. LEMMA. Let P be a flat M(G)-M(H)-bispace, and Q a flat M(H)-M(K)-bispace. S Then P®M(H)Q * a flat M(G)-M(K)-bispace, and there is a natural isomorphism of functors BK-^BG: = ~ ® M(G) (P ®M(//) Q)- ^i- ®M(C) P) ®M(H) Q- 2-5. Remark. The preceding lemma says that the functors t: Flat (M{G),M(H)) -> Homy (£#, £6?) send tensor products to compositions of geometric morphisms, i.e. that the square txt Flat (M(G), M(H)) x Flat (M(H), M(K)) • Homy(BH, BG) x Homy(BK, BH) Flat (M{G), M(K)) *• Homy(BG, BK) of categories and functors commutes up to natural isomorphism. Morita equivalence for continuous groups 105 Proof of 24. It is easy to see that P ®M(H) Q is an open bispace, i.e. that the three maps P ®mH) £-> 1, M(G) x (P ®M(H) Q) ^P ®M(H) Q and (P ®M(H) Q) xM(K) -^-» P ®M{H)Q are a^ °Pen- To see that the diagonal action map (cf. 2-1) M(G)xM(G)x(P®mH)Q)- Q) x •>M(H) Q) (i) mH) is open, consider the pullback R -M(O) x M(G) x P P.b. (2) PxM(H)xM(H) P *PxP, where ntoa is the composite M(G) xM(G) xP-^->M(G) xP—'—>P, and similarly for/?. Now a is open by hypothesis (this is part of the definition of'flat'), so \jf is open. Consider the diagram M(G) x M{G) xRxQ X M(G) X P) X (M(H) X M{H) X Q) (3) M(G) xM(G)xM(G)xM(G)xPx Q (PxP)x(QxQ) (IXI)OT (P® Q) x (P® Q) M(G)xM(G)x< 'M(H) Q) mH) M(H) where n is the quotient map P xQ^-P ®M So n is open, if we can show that (m x 77)0 (1 x 1 XI M(G) x M(G) xPxQ — M(G) x M(G) x (P®mmQ) where, writing 1—e—+M(G) and 1—e—+M(H) for the units, P—?-+R is given by ® M{G) =S ®M(G) (P ®M(H) Q) 106 IEKE MOERDIJK of H-sets, for each G-set S. Writing S as the sum of its orbits, it suffices to show that for each open subgroup U £ G, , Q)/Nv £ (P/Nv) ®MW Q (4) (cf. 2-2). Consider the diagram t PxM(H)xQ * P/Nvx M(H)xQ II PxQ P/NvxQ (5) (NuxNuxP)®M(H)Q =Z P®mH)Q " Here the rows come from applying —x(M(H)xQ), — xQ and — ®M(H)Q respectively to the stable coequalizer (3) in 21; the middle vertical one is (1) above 2-4, and the left-hand vertical coequalizer is isomorphic to the result of applying NJJ xN\j x — to the coequalizer (1) above 2*4, since this coequalizer is stable: (Nv xNvxP) ®mH)Q^(NvxNv) x (P®M{H)Q), (6) Taking first the coequalizers of the two upper rows, then of the last column, gives the same result as first taking the two left-hand coequalizers, and then taking the coequalizer along the bottom. In other words, the composition of the bottom and the isomorphism (6) is a coequalizer, i.e. (P ® M<«) Q)INV = (P/Nv) ®mm Q as was to be shown. 3. Geometric morphisms Let G and H be continuous groups. We will now show that each geometric morphism BH • BG is induced by a flat M(G)-i/(//)-bispace, as in 2-3. Given such a Write P-^-KJ>*(G/U) for the projection; each TIV is an open surjection ([10], 5-1 (ii)). Each PxM(H) *P is an open map. (1) M(G) acts on P from the left, M(G)xP—'—*P. This action is described in point-set notation by na(x-p) = WG/zJ Uxu -2L> G/U) (px-r UX(j), (2) where x = {x^yeiimG/U = M(G), and p = {p^u is a point of P. (2) can be taken as a definition of the map M(G) xP^-P, by using test-spaces as in 15. It follows Morita equivalence for continuous groups 107 easily from functoriality of 3-1. LEMMA. Let BH >BG be a geometric morphism, and P = lim (j>*(G/U) as above. Then P has the structure of an M(G)-M(H)-space. v We list some of the properties of P. Note that (ii) below together with (1) above imply that P is an open bispace. 3-2. LEMMA. Let BH-^BG and P be as in 31. (i) The action of M(G) on P is free, i.e. Avp M(G) x P >M(G) x M(G) x P is an equalizer, where /itj is M(G) xM(G) xP "" > M(G) '(ii) The action is transitive, in the sense that M(G) xM{G) xP(*s^5)PxP is an open surjection. Moreover: (iii) For any open subgroup U £ Cr, the horizontal map in NvxNvxP As a preparation for the proof of 32, let us observe the following. The 'point-set m Wl = interpretation' of (iii) is roughly that for all p = {jpu)v, q = {qv)u P ^ Pu Qu> there are x,yeNu £ M(G) and an rsP with rv = pv = qv such that x-r = p, and yr = q. Now consider the following special case of (iii), which contains the key construction. 3-3. LEMMA. Let BH >BG and P = \\m Proof. Assume for the moment that Sf = Sets, and that G has enough points. We may suppose that U = Uo. By induction, we will define a strictly increasing 108 IEKE MOERDIJK sequence 0 = n0 < w1 < ... of natural numbers, sequences (xk)keN, (2/fc)fceN of points of G, and a sequence (rnk)k, rnte$*(G/Unk), such that xk, ykeU, (0) (1) (2) rnt, (3) k, (4) V Then if we define rm = i>*(G/Unk G/Um)(rnk) when nk_± < m < nk, we obtain points x,y$M(G) and reP such that Put n0 = 0, a;0 = y0 = 1, rBo = p0 = q0. Suppose nk, xk, yk, rnt have been defined. By successive applications of flatness of fi*: £f(G) ->Sets, we construct a commutative diagram from right to left, as follows: G/Vk / G/Unt G/W'Q G/Uk Since and 4 Similarly, there are c,deG and W ^ d^U^dOc^U^c and a £efi*(G/W), such that ^*(Gr/^'^^G/f7nfc)(C) = rnk, 0*{G/W'-^G/Uk){Q = ?t. By the same argument, we find e,feG and V £ «rxWt O/"1^'/, and an i)e 1 Write ^ = (fte)" , let nk+l be so large that Morita equivalence for continuous groups 109 and let rnk+s Proof of 3-2. (i) is easy and left to the reader, (ii) follows from (iii). To prove (iii), it is not hard to see that it suffices to show that for any base extension $ • Sf and any points p, q of e*(P) with ^(p) = nv(q), there is an open surjection & • & such that in 2F there are points x, y of fi*e*(M((?)) and r of p*e*(P) such that in IF one has x • r = p and yr = q. But this follows from 33, because given 8——*y and p,q, we can start with an open surjective base extension &' —y-^S such that in &', G (or rather y*e*(G)) has a countable co-final system of open subgroups, by lemma 5-3 of [10]. 3-4. PROPOSITION. Let G and H be continuous groups, and let BH +BG be a geometric morphism. Construct the M(G)-M(H)-bispace P = lim Proof. It follows from 3-1 and 3-2 that both actions M(G) xP->P and P x M(H) H> P are open, and that the diagonal action M{G)xM(G)x.P->Px.P is open. Since both (j)* and —®M<,G)P preserve co-limits, it is enough to show that there is an isomorphism of fl-sets, ^(G/^-^-^P/NJJ, natural in G/V. (Recall from 21, 2-2 that P/NJJ is discrete, and that this is a stable quotient.) Write nv for the projection, and qu for the quotient map as in the diagram below. Clearly, nv factors through qv, say by TV, and TV is an open surjection since nv and qv are. It also follows from 32(iii) that TV is mono. Thus TV is an isomorphism. If (j> and ft are geometric morphisms BH^-BG and 3-5. LEMMA. The functor I: Homy (BG, BH) -> Flat (M(G),M(H)) is fully faithful. Proof. If a:0*->-^* is a natural transformation, the map l(a): l( I "V commute, so clearly I is faithful. On the other hand, if l( M(G), so induces a map l( U £ G. The map gv are the components of a natural transformation \<*u so I is full. 4. Torsors Let G and H be continuous groups, and let P be a flat M(G)-M(H)-\>\&ipa.ce, as in Section 2. Let P = UmP/Nv, where P/Nv is the quotient of P by the action ofNv £ M(G), cf. 2-2, and the inverse limit is taken over all open subgroups U c G, ordered by inclusion. Write for the canonical map. P is an M(G)-M(//)-space, and 0 is a map of bispaces. 41. LEMMA. For each open subgroup U £ G, 6 induces an isomorphism (1) Proof. We proved this already: let 1X0 The rows are coequalizers, and qvodo- = qvo6on2, so there is a unique a with fiqvd = TTU6 = qu, so ficx = id since qu is epi. On the other hand, 4-2. DEFINITION. A flat M(G)-M(H)-bispa,ce is called an M{G)-M(H)-torsor if P >P is an isomorphism. (Notice that by definition, P is a torsor iff P S U(P).) We Wnte Tor (if (G), if (#)), for the full subcategory of Flat (M(G),M(H)) consisting of torsors. Clearly, Tor (M(G),M(H)) is a reflective subcategory of Flat (if (G), M(H)): if P is a flat if (G)-if (#)-bispace, and T is an if (G)-if (#)-torsor, then for any map P-^T of bispaces there is a unique a: P-+T with ao#p = a. If PeTor (if (G),if(#)) and #e Tor (if (#), if (#)), we write An M(G)-M(H)-torsor is called invertible if there exists an M(H)-M(G)-torsor such that there are isomorphisms of bispaces The isomorphism classes of invertible M(G)-M(G)-torsors form a group under the operation —®M<,G)~> analogous to the Pi card group of a commutative ring. This group is denoted by Pic (G). 4-3. THEOREM. Let G and H be continuous groups. There is an equivalence of categories (natural in G and H) between geometric morphisms BH^-BG and M(G)- M(H)-torsors: Proof. The equivalence is given by the functors Homy (BH, BG) ^t Tor (M(G),M(H)). t Indeed, I obviously maps into torsors (cf. 34) and is fully faithful (35). That lot is isomorphic to the identity follows immediately from the construction, and Definition 4-2. One may collect the torsors for different groups G and H in one large (bi-)category Tor: the objects are the continuous groups, the morphisms (1-cells) from G to H are M(G)-M(H)-toTSOT8, and the 2-cells are maps of bispaces, i.e. Horn (G, H): = Tor (M(G),M(H)) as categories. Composition is given by tensor product (cf. 2-3), followed by reflection: Tor (M(G),M(H)) x Tor (M(H), M{K)) ^5 Tor (M (G), M (K)). (Recall that composition in a bicategory is only associative up to a coherent isomorphism, cf. [3].) 112 IEKE MOERDIJK 4-4. THEOREM. The category of atomic connected toposes with a point (over an arbitrary base topos £f) is dual to the category Tor^. of continuous groups and torsors in Proof. This follows from 1-9, 2-4 (and 2-5) and 43. If $ is an 5^-topos, Aut^,( S. 4-5. COROLLARY. There is a natural isomorphism of groups for any continuous group G in £f'. 4-6. COROLLARY ('Morita equivalence'). Let G and H be continuous groups. The following statements are equivalent : (1) BG and BH are equivalent categories. (2) There are an invertible M(G)-M(H)-torsor P and an invertible M(H)-M(G)-torsor Q such that (as bispaces) (3) There are a flat M(G)-M(H)-bispace P and a flat M(H)-M(G)-bispace Q such that Let 2F >$ be a geometric morphism. Then / induces a functor /*: (• (^"-spaces) from internal spaces in $ to internal spaces in SF; if X is a space in $, f*(X) is the space in 3F making 3F * g into a pullback. /* has a left-adjoint, so it preserves all projective limits ([7, 9]); moreover, it sends open maps to open maps. Since the quotients involved in the definition of flat bispace (2-3) and torsor (4-2) are stable (12), as is the construction of M(G) (1-4), it follows that/* induces two functors /*: Flat,(ilf( f*:Tore(M(G),M(H))^1otr(M(f*(G)),M(f*(H)))t where Flatg(M(G),M(H)) is the category of internal flat bispaces in &, for continuous groups G and H in S, etc. On the other hand, S' > Hom^(f x^.^x^) S S by pullback, for & -toposes 9C and °H'. Since everything involved in Theorem 4-3 is stable, we conclude: Morita equivalence for continuous groups 113 4-7. PROPOSITION. Let !F >& be a geometric morphism, and let G and H be continuous groups in S. Then 5 , H), B(S, G)) -^-••Honv(JB(J '1/*#), B(F,f*G)) Tore(M(G), M(H)) -^— Tor?(M(f*G), M(f*H)) commutes up to isomorphism (modulo the equivalence B(!F,f*H) ~ !F x@i($,H), cf. 1-8). # If G is a continuous group in a topos S, we write Torg(M(G)) for Torg(M(G),M(l)), where 1 is the trivial group. 4-8. COROLLARY ('BG classifies Jf(G)-torsors'). Let G be a continuous group in a topos S be a geometric morphism. There is a natural equivalence between geometric morphisms #"-> B(S,G) over S and internal M(f*G)-torsors in 2F: ^,B(S,G))-^-^Tor3 5. Pointed maps Let G and H be continuous groups. BG and BH have canonical points pG: £f -> BG f andpH:S/'^BE (1-7). A pointed map BH->BG is a pair (/,a), where BH — —>BG is a geometric morphism over Sf and a: pG ^-fpH is a 2-isomorphism. If (/, a) and (g,{l) are such pointed maps, a 2-cell (f,a)=>(g,fi) is a 2-cell u:f=>g such that (u-pH)oa = fi. This defines a category Notice that since p J is faithful, there can be at most one 2-cell u between two objects (/, a) and (g,P), and that u must be an isomorphism. So the 2-cells define an equivalence relation on Homy(BH,BG).. Let [BH,BG]% be the set of equivalence classes. 5-1. THEOREM. Let G and H be continuous groups. There is a natural isomorphism between equivalence classes of pointed geometric morphisms BH^-BG and continuous homomorphisms M(G) -+M(H): Horn (M{G),M(H)) -=*-+ [BH, BG].. f Proof. Let (/,a) be a pointed geometric morphism, where BH *BG and a: PG =>/PH/'S induced by an M(G)-M (H)-tovsov P, and a. gives a natural transforma- tion p*^~®M(G)P> w^h components So for each open subgroup U ^G,ocu: G/U ~>P/iV't/, and this gives an isomorphism a, of M(G)-spaces lim (?/£/-> lim P/Nv, i.e. a: M(G)-+P. 114 IEKE MOERDIJK Since M(H) acts on P from the right, there is a unique 8a: M(H)^-M(G) such that (ail),id) M(H) -PxM(H) I. (1) M(G) —- P commutes (where a(l) is the map 1 >M(G) " >P; so in point-set notation, 8a is defined by 8a{y)-a(l) = a(l)-y, or equivalently oc(8a(y)) = a(l)-y). It is easy to see that 8a is a homomorphism of continuous monoids. Now suppose u: (/, a.) ~ > (g, /?) is a 2-cell. If/ is induced by the torsor-P and g by the torsor Q, and M(G)——>P,M{G) *Q are the corresponding isomorphisms of M{G) -spaces, then u induces a map u of bispaces such that (2) commutes. Since u is in particular a map of right M(//)-spaces, it easily follows from commutativity of (1) and (2) that 8a = 8p. Conversely, a continuous homomorphism 8: M(H)->M(G) obviously makes M(G) into a right M(//)-space, and hence into an M(G)-M(H)-torsov. As a corollary, we obtain a result from [11]. 5-2. COROLLARY. Let G be a prodiscrete group, and H an arbitrary continuous group. Then there is a natural isomorphism Horn (H, G) ^-> [BH, BG].. Proof. If G is prodiscrete, then dG:G->M(G) is an isomorphism (cf. [11]). Since 6H: H->M(H) is dense, it induces an isomorphism Horn(M(H),G)-^-+ Horn(H,G). The result now follows from 51. One may also consider the full subcategory of Homy(BH, BG) whose objects come from Homy(BH,BG)m, i.e. the full subcategory of Uom^BH, BG)t consisting of those BH >BG for which there exists some unspecified isomorphism fpH =pG. Denote this category by Horny(BH, BG)+. The set of continuous homomorphisms M(H) *M(G) can also be made into a category: the maps ^=>v'r are points g: 1^-M(G) such that the identity 5*3. COROLLARY. Let G and H be continuous groups. There is a natural equivalence of categories Horn (M(H),M(G))-^-*Homy(BH, BG)+. Proof (sketch). By (the proof of) 51, the objects of Homy(BH,BG)m are precisely those torsors P for which there is an isomorphism M(G) " *P of M(G) -spaces. If M(G) —^P and M(G) -?-> Q are two such (with corresponding 8a, 80: M(G) -+M{H)), a 2-cell P >Q no longer makes the triangle (2) in Theorem 51 commute. Let Morita equivalence for continuous groups 115 g = fi~lua(i)\ 1^-M(G). It is easy to check that (in point-set notation) Sa(x)-g = g-8p(x). 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